Stochastic symplectic and multi-symplectic methods for nonlinear Schrödinger equation with white noise dispersion

Cui, Jianbo; Hong, Jialin; Liu, Zhihui; Zhou, Weien

doi:10.1016/j.jcp.2017.04.029

Cited by 30 publications

(19 citation statements)

References 20 publications

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“…Many stochastic differential equations, such as the stochastic oscillator, have stochastic Hamiltonian formulation and an associated stochastic symplectic structure. Concerning their numerical integration, stochastic symplectic methods have received extensive attentions (see e.g., [1,2,4,3,5,8,11,17,18,19,27,25,26] and references therein), for their superiority in numerical computations compared with non-symplectic ones. The motivation of this paper is to explain the superiority of stochastic symplectic methods, by studying the LDPs of numerical methods for a linear stochastic oscillator Ẍt + X t = α Ẇt with α > 0, and W t being a 1-dimensional standard Brownian motion defined on a complete filtered probability space (Ω, F , {F t } t≥0 , P).…”

Section: Introductionmentioning

confidence: 99%

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Chen,

Hong,

Jin

et al. 2019

Preprint

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It is well known that symplectic methods have been rigorously shown to be superior to non-symplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we focus on the superiority of stochastic symplectic methods applied to a linear stochastic oscillator, from the perspective of large deviations principle (LDP). Based on the Gärtner-Ellis theorem, we first study the LDPs of the mean position and the mean velocity for both the exact solution of the stochastic oscillator and its numerical approximations. Then, by giving the conditions which make numerical methods have first order convergence in mean-square sense, we prove that symplectic methods asymptotically preserve these two LDPs, in the sense that the modified rate functions of symplectic methods converge to the rate functions of exact solution. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the "hitting probability" of the mean position and mean velocity of the original system. However, it is shown that non-symplectic methods do not asymptotically preserve these two LDPs by using the tail estimation of Gaussian random variables. To the best of our knowledge, this is the first result about using LDP to show the superiority of stochastic symplectic methods compared with non-symplectic ones.

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Section: Introductionmentioning

confidence: 99%

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Chen,

Hong,

Jin

et al. 2019

Preprint

Self Cite

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“…which becomes an alternative to the approximations (11) and (18). It is worth pointing out that (19) becomes (18) [55]), using (10), (12) and (15) we construct the next numerical method for (1).…”

Section: Variants Of the Euler-exponential Schemementioning

confidence: 99%

Numerical solution of stochastic quantum master equations using stochastic interacting wave functions

Mora¹,

Fernández

Biscay

2018

Journal of Computational Physics

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We develop a new approach for solving stochastic quantum master equations with mixed initial states. First, we obtain that the solution of the jump-diffusion stochastic master equation is represented by a mixture of pure states satisfying a system of stochastic differential equations of Schrödinger type. Then, we design three exponential schemes for these coupled stochastic Schrödinger equations, which are driven by Brownian motions and jump processes. Hence, we have constructed efficient numerical methods for the stochastic master equations based on quantum trajectories. The good performance of the new numerical integrators is illustrated by simulations of two quantum measurement processes. t 0 Tr R m (s) * R m (s) ρ s− ds (see , e.g., [12, 31, 46]) such that N 1 , . . . , N M have no common jumps, and

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“…A review on stochastic multisymplectic schemes for Maxwell equations was given by Zhang et al in [5]. Cui et al [17] presented stochastic multisymplectic method to solve random Schrödinger equation. Hong et al [6,7] proposed multisymplectic methods and investigated discrete conservative quantity to stochastic Maxwell equations driven by additive noise.…”

Section: Introductionmentioning

confidence: 99%

Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions

et al. 2021

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Aiming at conservative Maxwell equations with periodic oscillatory solutions, we adopt exponentially fitted trapezoidal scheme to approximate the temporal and spatial derivatives. The scheme is a multisymplectic scheme. Under periodic boundary condition, the scheme satisfies two discrete energy conservation laws. The scheme also preserves two discrete divergences. To reduce computation cost, we split the original Maxwell equations into three local one-dimension (LOD) Maxwell equations. Then exponentially fitted trapezoidal scheme, applied to the resulted LOD equations, generates LOD multisymplectic scheme. We prove the unconditional stability and convergence of the LOD multisymplectic scheme. Convergence of numerical dispersion relation is also analyzed. At last, we present two numerical examples with periodic oscillatory solutions to confirm the theoretical analysis. Numerical results indicate that the LOD multisymplectic scheme is efficient, stable and conservative in solving conservative Maxwell equations with oscillatory solutions. In addition, to one-dimension Maxwell equations, we apply least square method and LOD multisymplectic scheme to fit the electric permittivity by using exact solution disturbed with small random errors as measured data. Numerical results of parameter inversion fit well with measured data, which shows that least square method combined with LOD multisymplectic scheme is efficient to estimate the model parameter under small random disturbance.

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Stochastic symplectic and multi-symplectic methods for nonlinear Schrödinger equation with white noise dispersion

Cited by 30 publications

References 20 publications

The probabilistic superiority of stochastic symplectic methods via large deviations principles

The probabilistic superiority of stochastic symplectic methods via large deviations principles

Numerical solution of stochastic quantum master equations using stochastic interacting wave functions

Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions

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